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I want to solve one system of ODE problem with numerical methods. I used RK2, RK3, RK4 and the Adams-Bashforth method. Now I want to solve it with higher-order Runge-Kutta methods. Do you have these tables? And do you know other explicit method to solve this problem?

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There are plenty options, see for instance the List on Wikipedia.

Examples for fifth order explicit schemes include Runge-Kutta Fehlberg and Dormand-Prince.

Yet another list can be found on the documentation pages of e.g. DifferentialEquations.jl or Matlab. You can probably find something similar for SciPy.

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